1. Impartial Games, Nim, Composite Games, Optimal Play in Impartial games
Basic Game Theory
Impartial Games, Nim, Composite Games, Optimal Play in Impartial games

2. Table of Contents (1) What is Game Theory?
Types of combinatorial games and plays
Impartial, Partisan
Normal & misère play
Game States
Turns; P, N positions
Optimal play
"Scoring" game example

3. Table of Contents (2) The Game Nim Composite games
Gameplay, winning positions
Composite games
Tweedledum & Tweedledee principle
Sprague-Grundy Theorem
Calculating Grundy Numbers
Sprague-Grundy function

4. What is Game Theory?

5. What is Game Theory? Study of decision making Mathematical models
Of conflict
Of cooperation
Between intelligent decision makers
Started with study of zero-sum games
One player winning leads to other player losing
Evolved into "decision theory"
General decision making strategies

6. Game Theory – Concepts Players, Actions, Payoffs, Information (PAPI)
Key concepts to describing a game
Solution strategy is based on PAPI
Solution strategy + PAPI = predictable, deterministic outcomes to games
Game theory fields of application
Politics, Economics
Phycology, Biology, Logic
All fields needing info on behavioral relations

7. Game Theory – Game Types
Notable general game classifications
Perfect vs. imperfect information
Cooperative vs. competitive
Symmetric vs. asymmetric
Combinatorial
Infinitely long
Discrete vs. continuous, differential, population, stochastic, metagames…

8. Impartial, Partisan and Play Types
Combinatorial Games
Impartial, Partisan and Play Types

9. Combinatorial Games Definition of a combinatorial game
Perfect information – players know at any time:
Game state
All possible player moves
No chance devices
Actions are not random and do not depend on random events
Two players move alternatively
No chance devices
Perfect information
The game must eventually end
Winner depends on who moves last – no draws

10. Partisan vs. Impartial Partisan games
Different moves available to different players
Chess, Checkers, Tic-Tac-Toe… are partisan
Impartial games
Different players have the same moves
Examples – Nim, Quarto
Impartial games can be generalized and analyzed
Very strong base for studying other games
Many partisan games, but with similar principles

11. Combinatorial Games Play types – determine which player wins
Based on who moves last
Normal play
Player which makes last possible move wins
E.g. player who takes the last coin wins
Misère play
Player which makes last possible move loses
Aim to force the other player into the last move

12. Game States & Positions
Turns, P and N Positions in Optimal Play

13. Game States – Chips Game “chips” (or “pieces”)
A chip is something a player controls
E.g. figures in chess, stack(s) of coins, numbers to operate on, etc.
Chips are always in some state (i.e. position)
A position can be winning or losing
In other words, good for next or for previous player

14. Game States - Turns Impartial games are played in turns
Positions are usually the "thinks" stage
Impartial game states – positions & count of chips
Blue
moves
Yellow
moves
Blue
moves
Blue
"thinks"
Yellow
"thinks"
Blue
"thinks"
Yellow
"thinks"

15. Game States – Player Position
Position of a player
Depends on the positions of all the chips
Considered in the "think" stage
Immediately after the other player moved
Immediately before the current player moves
Two types of positions
Good for current player (i.e. next to move)
Good for previous player (i.e. who just moved)
Usually noted P (previous) and N (next/current)

16. Game States – Optimal Play
Impartial games are deterministic
Optimal play always exists
Player position – either winning or losing
Optimal play in impartial games
Best actions by player, in order to win
Optimal play + winning position = victory
So, N and P positions for the current player:
N – Winning positions
P – Losing position

17. How to Play Optimally Impartial games are zero-sum
Positions are either winning or losing
Try to force other player into losing position
i.e. a P-position (good for previous player -> us)
If we are in a losing position, we can’t win
if other player plays optimally

18. Game of Scoring Scoring is a game where a single chip is moved
Right to left, starting at a numbered position
Moves have maximal step – e.g. 3 positions
Move chip by no less than 1 and no more than 3
0 (zero) is the leftmost position
Can’t move from that position
Player which cannot move is the loser

19. Game of Scoring – Demo Let’s try to determine P and N positions
Max step is 3 1 2 3 4 5 6 7 8 9 P N

20. Game of Scoring – Positions
0 – losing (P-position)
1, 2, 3 – winning positions (N-positions)
We can immediately move the chip to 0
4 – losing position (P-position)
We place chip at 1, 2 or 3 and other player wins
5, 6, 7 – winning positions (N-positions)
Can place other player in a losing position (4)
Scoring has a direct formula for P-positions

21. Generalization of Positions
For all impartial games
A position is winning if it can lead to at least one losing position
A position is losing, if it leads only to winning positions
This regards single-chip games
P-position
N-position
Every option leads to an N-position
There is always at least one option that leads to a P-position

22. Gameplay, Positions, Nimbers and Nim-sum
The Game Nim
Gameplay, Positions, Nimbers and Nim-sum

23. Nim Ancient, but fundamental game Rules Impartial, Many variations
Nim theory developed early 20th century
Important related terms – nimbers, Nim-sum
Rules
Several heaps (piles) of stuff (coins/stones/...)
Player takes 1 or more elements from 1 pile
Normal play – player taking last element wins

24. Nim – formal notation
Nim state is easy to express formally
If the number of heaps is k
Then (n0, n1, …, nk) is the state
n0 is the number of items in the first heap, etc.
State is often referred to as position
If we have 3 heaps of sizes 3, 4 and 2
Then we are at position (3, 4, 2)
Position (0) is losing
In normal play

25. Nim - Demo
Let's play Nim
index 1 2 3 4 5

26. Nim – Observations What happens for Nim with 1-element heaps?
(1) – obviously, first player wins (N)
(1, 1) – first player takes a heap, second wins (P)
(1, 1, 1) – first player forces second player to case b), so second player will lose (N)
(1, 1, 1, 1) – first player goes into case c), so second player will win (P)
Nim with k 1-element heaps
Winning position if k is odd

27. Nim – More Observations
What happens for Nim with 1-element heaps mixed with 2-element heaps?
(1, 2) – first player wins (winning position)
first player can reduce 2 to 1
player 2 forced into (1, 1), which is losing
(2, 2) – other player wins (losing position)
First player’s moves are to (1, 2) or (0, 2) *
First option leads second player to case a)
Second option – second player takes the heap

28. Nim – General Observations
We could go on generating positions
That would take too much (exponential) time
Another pattern can be noticed
Even same-sized heap count – bad position
Odd same-sized heap count – good position
b) + one larger heap – good position
can move to position a) by reducing larger heap
Other similar even-odd considerations
Can we easily filter through good/bad positions?

29. Nim-Sum Nim-sum (aka XOR, ^, addition modulo two)
The Nim-sum of a position (n0, …, nk) is n0 ^ n1 ^ … ^ nk
Accredited to Charles L. Bouton as part of the solution of the game
Mathematical notation typically: n0 ⊕ n1 ⊕ … ⊕ nk
Heap sizes are usually called nimbers
A non-zero Nim-sum denotes a winning position
A zero Nim-sum denotes a losing position

30. Nim-sum – Why it Works Position (0) has a Nim-sum = 0
No moves can be made – position is losing
Position (k) has a Nim-sum > (k > 0)
Player takes the entire heap – position is winning
Changes to a position with Nim-sum = 0
Always lead to positions with Nim-sum > 0
Changes to a position with Nim-sum > 0
At least one change leading to Nim-sum = 0
So, we can always force a losing position
From a winning position

31. Nim-sum – Finding Positions
Considering we are in a winning position
Need to search for positions with Nim-sum zero
Finding a zero Nim-sum position
Decrease numbers, which make the Nim-sum > 0
Achieve even number of 1’s in each bit column
Calculate the current Nim-sum
Finds its leftmost bit equal to 1
(denotes a column with an odd number of 1’s)
Find any number N (heap size) having a 1 at that bit
Nullify that number N
Calculate the Nim-sum again
Set the nullified number N to the new Num-sum

32. Solving Nim in C#
Live Demo

33. Nim Importance What’s the big deal with Nim? Who?
The Sprague-Grundy theorem
Who?
R. P. Sprague & P. M. Grundy in 1935 & 1939
Independently discovered the theorem
The Sprague-Grundy theorem states:
I.e. every impartial game is a Nim in disguise or some variant of it
Every impartial game, under normal play is equivalent to a nimber

34. Nim Importance Consider the game Nimble
Several coins, at some positions
Must move exactly one coin at least 1 position
Coins move only right to left (towards zero)
Player who moves the last coin to 0 wins
Seem familiar? 1 2 3 4 5 6 7 8 9

35. Nim Importance How about now?
The solutions of Nimble and Nim are equivalent
After indices are turned into heap sizes
index 1 2 3 4 5

36. More General Nim?
In standard Nim, we can take any number of elements from 1 pile
We can generalize Nim for K piles
Take any number of elements from at least 1 pile and at most K piles
This is called Moore's Nim
More formally, for K piles it is Nimk
So normal Nim is actually Nim1
Can you think of a way to edit the Nim solution to work for Moore's Num?

37. More General Nim (Moore's Nim) – Solution
We need to change the Nim-sum operator
For Nim1 (Standard Nim) we convert piles to binary and use "sum modulo 2"
So for Nimk we again convert piles to binary, but instead use "sum modulo k+1"
E.g. if we can take from at most 2 piles, Nim-sum is "sum modulo 3"
The remaining part of the solution is the same
Positions with Nim-sum > 0 are winning
Positions with Nim-sum < 0 are losing

38. Composite Games
Dividing into Subgames, Tweedledum & Tweedledee Principle, Sprague-Grundy

39. Composite Games
Generally, composite games can be broken down into subgames
Usually composite games can be formed from any game
By adding more chips
Examples:
Game of Scoring with N Chips
Nim with N groups of piles
Etc.

40. Game of Scoring – Demo 2 Let’s play Scoring again
This time with 2 chips
Max step is 3
Are the marked P/N positions correct in this case? 1 2 3 4 5 6 7 8 9 P N

41. Composite Games – 2 Chips
Tweedledum & Tweedledee Principle
2-chip game, the second player can mimic moves
If the two chips are in the same position
Leading to victory for the second player
In positions which would be winning for the first player in a single- chip game
Losing positions remain losing, problem is with winning ones
Similar cases occur in games with more chips
And more equivalent positions

42. Composite Games – Solving
Composite impartial games
Equivalent to nimbers (as other impartial games)
A game has k chips, each with a position,
Game position – set of k numbers (n0, …, nk)
Equivalent to a position in Nim, hence
Nim-sum of the position = 0 -> position is losing
How do we get those numbers/nimbers?

43. Calculating the Sprague-Grundy Function
Follower positions,
Minimal excludent

44. Calculating SG function
The nimbers we needed to describe a composite game's state can be calculated with the Sprague Grundy function
Yep, those guys again
Sprague-Grundy (SG) function
Recursively generates "Grundy" values/numbers
Numbers correspond to heap sizes in Nim
SG function a has "helper" functions to adapt to a specific game
The "Follower" function

45. Calculating SG function
Sprague-Grundy function – formal description:
g – Sprague-Grundy function
F – follower function
Mex – minimal excludant
p and q – two game positions
g(p) = Mex(g(q)); q ∈ F(p)

46. Calculating SG function
Follower function F(position)
Returns positions, reachable in 1 move from given position
i.e. "followers" of that position
Minimal excludant Mex(numbers)
Returns the minimal non-negative number
Not belonging to a given set
i.e. first number different than any of the given numbers

47. Calculating SG function
So, this reads as follows:
The Grundy value of position p
Is the minimal non-negative integer
Which is NOT a Grundy value of
Any of the followers of position p
g(p) = Mex(g(q)); q ∈ F(p)

48. Calculating SG – example
Grundy values with Scoring
Losing positions in a game have g() = 0
g(0) = 0, by definition.
g(1) = 1 since g(F(1)) = {0}.
g(2) = 2 since g(F(2)) = {0, 1}.
g(3) = 3 since g(F(3)) = {0, 1, 2}.
g(4) = 0 since g(F(4)) = {1, 2, 3}.
- The values of g on F(4) are {1, 2, 3} and the minimum that does not appear there is 0.
g(5) = 1 since g(F(5)) = {2, 3, 4}.
- The values of g on F(5) are {2, 3, 0} and the minimum that does not appear there is 1...
Index 1 2 3 4 5 6 7 8 9
SG value

49. Calculating SG – pseudocode
Grundy values pseudocode
Walk positions recursively
Generate set of Grundy values for followers
Take the Mex of the Grundy values of the followers
int GrundyNumber(position pos) {
moves[] = Followers(pos);
set s;
for (all x in moves)
s.insert(GrundyNumber(x)); }
//Mex – return smallest non-zero //integer, not in s;
int mexCandidate=0;
while (s.contains(mexCandidate))
mexCandidate++;
return mexCandidate;

50. Solving Composite Games Using SG & Nim
Determining a position as winning or losing
We have the positions of the chips in the game
We have the Grundy values of those positions
Hence, we have the Nim position:
(g(p0), g(p1), … g(pk))
Where p0 … p1 are the chip positions
So, we can compute the Nim-sum
If it is zero, we are in a losing position
If it is non-zero, we are in a winning position

51. Solving 2-Chip Scoring with Sprague-Grundy and Nim
Say we have the following position, with SG numbers below
We have to 2 chips at position 7 and g(7) = 3
So, this is equivalent to a Nim with heaps (g(7), g(7)) = (3, 3)
which is a losing position because 3 xor 3 = 0 1 2 3 4 5 6 7 8 9

52. More Game Examples
And their solutions

53. Northcott's Game – (click to play on cut-the-knot.org)
Rectangular board, 2 checkers per row, 1 per player
Move rule: move exactly 1 checker anywhere on it's row, without jumping over an opponent's checker
Lose rule: first player which cannot move loses
Video: Solution of a task with this Telerik Algo Academy 1 2 3 4 5 6 7 8 9 10 11

54. Corner the Lady – (click solution on cut-the-knot.org)
Rectangular board, 1 checker, players take turns moving
Move rule: move left, down or left-down diagonally by any amount of cells. You can't move from the bottom-left position (1, 1)
Lose rule: first player which cannot move loses 9 8 7 6 5 4 3 2 1
* This is actually a "visual adaptation" version of Wythoff's Nim – read the description at the cut-the-knot link above

55. Other Impartial Combinatorial Games
Northcott's Game with moving chips in more than 1 row
Spolier: This is a variant of Moore's Nim
Corner the Lady with more than 1 chip
Spoiler: Composite game – calculate SG values and Nim-sum to solve
Dots and Boxes, with "first player to make a box wins" rule
See this article on composite mathematical games at Oxford
The Knights game in this TopCoder article
A lot of games on

56. Conclusion What we covered Impartial games
Winning and losing positions (N and P)
Nim and representing impartial games as Nim
Sprague-Grundy theorem & function
Solving composite games

57. Conclusion What we haven’t covered
Misère play
Partisan games
Non-deterministic games
The above share some characteristics with impartial games
Most games are describable by mathematical models, which are based on similar concepts to those in the lecture. Just take your time to unravel the model – here, programming is the easy part once you’ve understood the logical base of the problem.

58. Questions?